Hilbert–Burch theorem

In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968,p.944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof.


If R is a local ring with an ideal I and

is a free resolution of the R-module R/I, then m = n  1 and the ideal I is aJ where a is a non zero divisor of R and J is the depth 2 ideal generated by the determinants of the minors of size m of the matrix of the map from Rm to Rn.


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